NEW HAVEN, Conn.—William Walker waited on an ice-covered sidewalk for the two police officers who usually walk the beat in his neighborhood. He wanted to report a car that didn’t belong there but didn’t want to make a big deal out of it.

“You have somebody walking around, you can talk to them,” says Mr. Walker, 44 years old, who works at a local hospital. “They don’t come in the neighborhood and treat everybody like they’re the enemy.”